Part of the confusion might be that a “group” in Blender is not like in other applications, and should probably have been called something else—a “grouped selec

Part of the confusion might be that a “group” in Blender is not like in other applications, and should probably have been called something else—a “grouped selection”, say. In Blender terms, you probably wanted to create a “parent” (typically an empty) for your objects.